Editing Tensor algebra (section)

==Adjunction and universal property==
The tensor algebra {{math|''T''(''V'')}} is also called the '''[[free algebra]]''' on the vector space {{math|''V''}}, and is [[functor]]ial; this means that the map <math>V\mapsto T(V)</math> extends to [[linear map]]s for forming a ''functor'' from the [[category (mathematics)|category]] of {{mvar|K}}-vector spaces to the category of [[associative algebra]]s. Similarly with other [[free object|free constructions]], the functor {{math|''T''}} is [[left adjoint]] to the [[forgetful functor]]  that sends each associative {{math|''K''}}-algebra to its underlying vector space.

Explicitly, the tensor algebra satisfies the following [[universal property]], which formally expresses the statement that it is the most general algebra containing ''V'':
: Any [[linear map]] <math>f:V \to A</math> from {{math|''V''}} to an associative algebra {{math|''A''}} over {{math|''K''}} can be uniquely extended to an [[algebra homomorphism]] from {{math|''T''(''V'')}} to {{math|''A''}} as indicated by the following [[commutative diagram]]:

[[Image:TensorAlgebra-01.png|center|Universal property of the tensor algebra]]

Here {{math|''i''}} is the [[Inclusion map|canonical inclusion]] of {{math|''V''}} into {{math|''T''(''V'')}}. As for other universal properties, the tensor algebra {{math|''T''(''V'')}} can be defined as the unique algebra satisfying this property (specifically, it is unique [[up to]] a unique isomorphism), but this definition requires to prove that an object satisfying this property exists.

The above universal property implies that  {{mvar|''T''}} is a [[functor]] from  the [[category of vector spaces]] over {{math|''K''}}, to the category of {{math|''K''}}-algebras. This means that any linear map between {{math|''K''}}-vector spaces {{math|''U''}} and {{math|''W''}} extends uniquely to a {{math|''K''}}-algebra homomorphism from {{math|''T''(''U'')}} to {{math|''T''(''W'')}}.